22-Mec-B4 Integrated Manufacturing Systems · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, May 2017 — 16-Mec-B4 Integrated Manufacturing Systems. Three hours, open book, any non-communicating calculator permitted. Six questions are printed and any five constitute a complete paper; all questions are of equal value, so each is treated below as a 20-mark question. Every question is solved here, because the set is a study resource rather than a sitting.
Reference texts. M. P. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 5th ed. (process planning, CAPP, group technology, discrete control and programmable logic controllers); R. B. Chase and F. R. Jacobs, Operations and Supply Chain Management, 16th ed. (forecasting); S. Nahmias and T. L. Olsen, Production and Operations Analysis, 7th ed. (inventory models and lot sizing); C. E. Ebeling, An Introduction to Reliability and Maintainability Engineering, 3rd ed. (series and parallel reliability); S. Kalpakjian and S. R. Schmid, Manufacturing Engineering and Technology, 8th ed. (machinability data and cutting conditions).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
The classical economic order quantity, $Q^{*}=\sqrt{2DS/H}$, rests on four assumptions that are easy to forget: demand is known, constant and continuous; each item is ordered independently of every other; replenishment is a single instantaneous delivery; and the whole cost of ordering is incurred once per order, per item. Each of the three situations in this question breaks a different one of those assumptions, and the right answer in each case is to name the assumption that fails and to move to the model that relaxes it. Applying the textbook square-root formula item by item in any of the three would give an answer that is arithmetically correct and operationally wrong.
(a) Fifty items shipped weekly to a branch warehouse. What fails here is the independence assumption. The cost that matters is the cost of despatching a shipment, not the cost of adding one more line to it, so the fifty items are not fifty separate inventory problems — they are one joint replenishment problem. The correct approach is a coordinated, or joint, order policy: choose a common replenishment interval $T$ for the group and, once the interval is fixed, order each item up to a target level rather than in a fixed quantity. If $S$ is the cost of raising and shipping one consignment and $s_i$ the small incremental cost of adding item $i$ to it, the economic order interval that minimises the group's total cost is $$T^{*}=\sqrt{\frac{2\left(S+\sum_i s_i\right)}{\sum_i D_i H_i}},$$ and the quantity of each item is then simply $Q_i = D_i T^{*}$. Items with very different turnover can be assigned integer multiples of $T^{*}$ — a power-of-two policy — so that a slow mover is replenished every second or fourth shipment while the base cycle stays weekly. In this particular case the shipping frequency is already fixed at one week by the transport arrangement, so the real decision is not the interval at all but the order-up-to level for each item, $S_i=\bar{d_i}\,(T+L)+z\sigma_i\sqrt{T+L}$, where $L$ is the transit lead time and $z$ sets the branch service level. The proper use of the EOQ formula here is as a check: compute $T^{*}$, and if it comes out near one week the weekly shipment is already economic, while if it comes out at three weeks the shipping schedule itself should be re-examined.
(b) A highly seasonal item. What fails here is the constant demand rate. The EOQ derivation averages demand over the year, and for a seasonal item that average describes no month at all: ordering the annual $Q^{*}$ in the trough leaves stock sitting for months, and ordering it in the peak leaves the item short. The approach is to abandon the continuous model and work from a time-phased requirements schedule, then apply a dynamic lot-sizing rule to it. Wagner–Whitin dynamic programming gives the optimal answer for a finite horizon; in practice the heuristics are used, either the Silver–Meal least-period-cost rule, which extends the cover period while the average cost per period keeps falling, or part-period balancing, which extends it while accumulated part-periods stay nearest the economic part-period $S/H$. If the item is a genuine one-season product with no carry-over — seasonal apparel, a promotional item — the problem is not a lot-sizing problem at all but a single-period newsvendor buy, and the order quantity is set from the critical ratio $C_u/(C_u+C_o)$ of the demand distribution. A defensible middle course, where seasonality is smooth rather than violent, is to keep the EOQ form but recompute it on a rolling basis with $D$ set to the current local demand rate, $Q_t=\sqrt{2D_t S/H}$, provided the resulting cycle is short compared with the season.
(c) The casting machined, finished and consumed on an assembly line. Two assumptions fail at once. The demand at every stage except the assembly line is dependent — it is derived from the schedule downstream, not observed independently — and the replenishment at the machining stages is not an instantaneous delivery but a production run at a finite rate. This is a multi-echelon, dependent-demand problem, and the governing approach is material requirements planning driven by the assembly schedule, with a lot-sizing rule applied stage by stage against echelon holding costs rather than full stage costs. Four separate EOQs, one per inventory point, would be the classic error: each stage would choose its own convenient cycle, the cycles would not nest, and semi-finished stock would pile up between them. Within that framework each stage takes the model that fits it. The casting is a genuine purchase with a supplier set-up or minimum-buy, so a true EOQ, extended to a quantity-discount analysis if the foundry offers price breaks. The chucking, milling and grinding operations are production lots on a machine that also runs other work, so the economic production quantity applies, $$Q^{*}=\sqrt{\frac{2DS}{H\left(1-d/p\right)}},$$ with $p$ the production rate and $d$ the usage rate; the factor $\left(1-d/p\right)$ recognises that stock builds only at the net rate while the run is in progress. Finally, because the assembly line withdraws continuously, the finished-component inventory should be small and pulled: a kanban loop between grinding and assembly, sized from the container quantity and the replenishment lead time, is the correct policy there. The lot sizes at successive stages should be chosen as integer multiples of one another so the stages stay synchronised, which is the multi-echelon result that a stage-by-stage EOQ cannot produce.
The unifying point is worth stating plainly for a marker. EOQ is not a formula to be applied, it is a trade-off between a fixed charge and a carrying charge, and every one of these three cases keeps the trade-off while changing what the fixed charge attaches to: a shipment in (a), a period in (b), and a production set-up inside a dependent-demand chain in (c). Identify the fixed charge correctly and the right model follows.
